Alexandrov topologies are the topologies induced by relations. This paper addresses the properties of Alexandrov topologies as the extensions of strong topologies and strong cotopologies in complete residuated lattices. With the concepts of Zhang’s completeness, the notions are discussed as extensions of interior and closure operators in a sense as Pawlak’s the rough set theory. It is shown that interior operators are meet preserving maps and closure operators are join preserving maps in the perspective of Zhang’s definition.

Pawlak [1, 2] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. Hájek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. By using the concepts of lower and upper approximation operators, information systems and decision rules are investigated in complete residuated lattices [3-7]. Zhang and Fan [8] and Zhang et al. [9] introduced the fuzzy complete lattice which is defined by join and meet on fuzzy partially ordered sets. Alexandrov topologies [7, 10-12] were introduced the extensions of fuzzy topology and strong topology [13].

In this paper, we investigate the properties of Alexandrov topologies as the extensions of strong topologies and strong cotopologies in complete residuated lattices. Moreover, we study the notions as extensions of interior and closure operators. We give their examples.

Definition 1.1.[3, 4] An algebra (L, ∧, ∨, ⊙, →, ⊥, 𝖳) is called a complete residuated lattice if it satisfies the following conditions:

Definition 1.7.[8, 9] Let (X, eX) be a fuzzy partially ordered set and A ∈ LX.

(1) A point x0 is called a join of A, denoted by x0 = ⊔A if it satisfies (J1) A(x) ≤ eX(x, x0), (J2) . A point x1 is called a meet of A, denoted by x1 = ⊓A, if it satisfies (M1) A(x) ≤ eX(x1, x), (M2) .

Remark 1.8.[8, 9] Let (X, eX) be a fuzzy partially ordered set and A ∈ LX.

(1) x0 is a join of A iff. (2) x1 is a meet of A iff . (3) If x0 is a join of A, then it is unique because eX(x0, y) = eX(y0, y) for all y ∈ X, put y = x0 or y = y0, then eX(x0, y0) = eX(y0, x0) = 𝖳 implies x0 = y0. Similarly, if a meet of A exist, then it is unique.

The fuzzy complete lattice is defined with join and meet operators on fuzzy partially ordered sets. Alexandrov topologies are the extensions of fuzzy topology and strong topology.

Several properties of join and meet operators induced by Alexandrov topologies in complete residuated lattices have been elicited and proved. In addition, with the concepts of Zhang’s completeness, some extensions of interior and closure operators are investigated in the sense of Pawlak’s rough set theory on complete residuated lattices. It is expected to find some interesting functorial relationships between Alexandrov topologies and two operators.